Post on 26-Jul-2020
A theory of operational cash holding, endogenousfinancial constraints, and credit rationing
Abstract
This paper develops a theory of operational cash holding. Liquidity shocks due to
delayed payments must be financed using cash or short-term debt. Debt holders
provide an irrevocable credit line given a firm’s expected insolvency risk, and eq-
uity holders select optimum cash holding. The model demonstrates the trade-off
between cash holding and investing in fixed assets. Introducing uncertain cash-
flows leads to precautionary cash holding if debt holders impose financial con-
straints. Precautionary cash holding, in turn, reduces insolvency risk enhancing
access to short-term finance. The theory shows that credit rationing can occur in
the absence of market frictions. Using U.S. data from 1998 to 2012, empirical
findings suggest that the decline in credit lines has contributed to the increase in
cash holding in line with theoretical predictions.
Keywords: cash holding, financial constraints, credit rationing, working capital
management
Preprint submitted to Working Paper March 29, 2016
1. Introduction
Cash ratios of U.S. companies more than doubled from 10.5% in 1980 to
23.2% in 2006 (Bates et al., 2009) and have remained on high levels through-
out the financial crisis. At the same time, recent discussions in the media stress
that firms do not have sufficient access to bank finance, undermining the economic
recovery. High levels of cash holding, credit rationing and lackluster investment
seem to coexist, which is hard to comprehend drawing on established theories.
For instance, investment opportunities should increase cash holding (Myers, 1977;
Myers and Majluf, 1984). A theory that aims to explain this phenomenon has to
derive optimum cash holding in the presence of endogenous financial constraints,
and there needs to be a link between cash holding and investment. To isolate
cash holding from other decisions such as capital structure, we focus on model-
ing short-term decisions during the cash conversion cycle (CCC) (Deloof, 2001;
Gitman, 1974; Richards and Laughlin, 1980). In a short period of a few days,
capital structure and dividends are fixed. This is consistent with Lins et al. (2010)
who argue that most of the increase in cash holding is due to operational cash
holding. Operational cash holding funds a firm’s daily activities and is part of
working capital management. Until recently, the literature has overlooked the im-
portance of working capital management. Jacobson and Schedvin (2015), Kling
et al. (2014) and Kieschnick et al. (2013) provide empirical evidence, but a theory
is missing. This paper proposes a theory of operational cash holding, endogenous
financial constraints, and credit rationing. The paper tests theoretical predictions
2
using U.S. data, uncovering the role of credit lines in understanding the increase
in operational cash holding.
Theories have determined the demand for cash based on four motives: trans-
action,1 precaution,2 investment opportunities, and self-interest.3 Acharya et al.
(2012) model the link between cash holding and credit risk, extending theories that
do not capture financial constraints explicitly. For instance, Gryglewicz (2011)
assumes constraint firms cannot raise additional capital after the initial stage, and
Almeida et al. (2004) use proxies to classify companies. The three-period model
developed by Acharya et al. (2012) derives precautionary cash holding in the pres-
ence of financial constraints, and Acharya et al. (2012) distinguish cash-flows
from existing assets (labeled xt), which are uncertain, and certain cash-flows from
an investment project that can be financed only by cash since market frictions re-
strict access to external finance. Cash-flow risk is additive in that flows in period
t = 1 refer to a known constant x1 plus a random shock u with E(u) = 0. Thus,
cash holding does not influence cash-flow risk. Acharya et al. (2012) derive pre-
cautionary cash holding as a buffer against negative cash-flows and a response to
financial constraints.
The first contribution is to derive operational cash holding caused by a firm’s
short-term liquidity need, summarized in Theorem 1. The model follows Holm-
1Baumol (1952); Miller and Orr (1966).2Keynes (1936) and recently Almeida et al. (2004); Gryglewicz (2011); Han and Qiu (2007);
Riddick and Whited (2009).3Self-interest leads to stockpiling of cash and value-destroying mergers (Graham and Harvey,
2001; Harford et al., 2008; Harford, 1999).
3
strom and Tirole (1997) and Holmstrom and Tirole (1998) among others who view
credit lines as pre-committed debt capacity. In the model, firms use credit lines to
finance short-term liquidity needs and not long-term projects, as Lins et al. (2010)
suggest. Banks often refer to ”working capital lines of credit” (TCF Financial Cor-
poration, Minnesota, USA), which best describes use of credit lines in the model.
As Acharya et al. (2013) argue on p. 1, ”Credit lines can be effective, and likely
cheaper substitutes for corporate cash holding,” and they are a substantial source
of liquidity for U.S. firms (Sufi, 2009; Yun, 2009). However, Acharya et al. (2013)
argue that revocations weaken commitment. In our model, firms cannot increase
liquidity risk directly, and hence there is no ’illiquidity transformation,’ which
justifies revocations (Acharya et al., 2013).
The second contribution is to make financial constraints endogenous and
demonstrate that firms can influence insolvency risk through cash holding (see
Theorem 2). Denis and Sibilkov (2010) and Han and Qiu (2007) stress the im-
portance of financial constraints, but established theories do not permit endoge-
nous financial constraints (Acharya et al., 2012; Almeida et al., 2004; Denis and
Sibilkov, 2010; Gryglewicz, 2011). In our model, cash holding has two effects:
(1) the ’buffer effect’ in line with Acharya et al. (2012) and (2) the risk-reduction
effect since cash holding reduces fixed assets, limiting exposure to cash-flow risk.
This finding reflects multiplicative instead of additive risk (Acharya et al., 2012).
Multiplicative risk does not allow use of simple diffusion of cash-flows (e.g.,
Brownian Motion) applied by Gryglewicz (2011) as cash-flows are partly en-
4
dogenous. Accordingly, the model focuses on a three-period context, which is
sufficient to derive primary findings and consistent with the short-term view.
The third contribution is to demonstrate that credit rationing occurs in the
absence of information asymmetry (see Theorem 3). Highlighted by Wolfson
(1996), the predominant view in the literature is that information asymmetry
causes credit rationing as shown by Stiglitz and Weiss (1981) and ’pure’ credit ra-
tioning models that followed. Alternative explanations refer to asymmetric expec-
tations (Wolfson, 1996), varying attitudes toward risk (Tobin, 1980), and covenant
violations (Sufi, 2009; Yun, 2009). This paper suggests that credit rationing occurs
in the absence of market frictions. Information is symmetric; expectations and at-
titudes toward risk do not differ among debt and equity holders. We contribute
to earlier work reviewed by Baltensperger (1978), which derives credit rationing
without resorting to market frictions (Jaffee and Modigliani, 1969).
2. The Model
2.1. Model structure and assumptions
Figure 1 illustrates the three-period structure with interim point, which fol-
lows Holmstrom and Tirole (1998) and Holmstrom and Tirole (2000). Our model
is set in a frictionless environment (e.g. no taxes), and we make the following
assumptions.
A-1 Equity and debt holders are risk-neutral.
A-2 Debt and equity holders are price-takers.
A-3 There are no agency costs.
5
A-4 The discount rate from t = 0 to t = 2 is zero.
A-5 Net working capital, w, is uniformly distributed with w ∼ U(w,w).
A-6 The cost-income ratio, k, is uniformly distributed with k ∼ U(0, k).
A-7 Net working capital, w, and the cost-income ratio, k, are independent.
A-8 Cash holding does not earn interest.
Debt holders behave like risk-averse agents due to their asymmetric payoff
structure and A-1. A-2 implies that modeling the market structure such as in Jaf-
fee and Modigliani (1969) extends beyond the scope of this paper. A-3 suggests
that managers and equity holders maximize shareholder value. A-4 simplifies our
model as we do not have to apply discount rates during the CCC. This is a plau-
sible assumption given our short-term perspective. A-6 is relevant for modeling
precautionary cash holding as we need to permit insolvency risk. A-7 simplifies
our model. Robustness checks in section 3 relax assumptions A-5, A-6 and A-7,
permitting a firm to exercise a degree of control over its net working capital, w.
Finally, A-8 ensures consistency in line with A-4. We express all variables rel-
ative to total assets. By doing so, we model cash ratios defined as cash holding
divided by total assets. This ensures that firm size (i.e. total assets A) does not
affect optimal cash ratios directly.
(Insert Figure 1)
At t=0, a new firm emerges with total assets, A, financed by equity and long-
term debt. Debt holders form expectations about the firm’s insolvency risk, π, and
determine an irrevocable credit line, s, for a given interest rate, r. Equity holders
6
observe the debt holder’s choice and select a cash ratio, c, knowing the firm’s pa-
rameters such as capital turnover, T , depreciation rate, l, long-term interest rates,
i, and financial leverage, L. Only the proportion of total assets invested in fixed
assets, 1−c, generates cash-flows. Fixed assets produce revenue according to cap-
ital turnover, T .4 The cost-income ratio, k, translates revenue into earnings before
depreciation, (1− c)T (1− k). We define costs only as operating costs (e.g. inputs,
labor) and exclude any costs related to net working capital (e.g. interest expenses
for using short-term finance). Fixed assets depreciate at rate l.
As the firm starts trading at t=0, there is no initial net working capital, w.
Equity holders are aware that an unknown liquidity need arises at t=1 as some
customers might not pay but some suppliers receive their payments. Hence, at
t=1, the firm experiences cash inflows, REV1, and outflows, COS 1. The mis-
match between outflows and inflows determines the short-term liquidity need
LIQ = COS 1 − REV1 or expressed in terms of total assets ν =LIQ
A , which re-
quires financing through cash holding, c, and short-term debt, s. Liquidity default
occurs if cash and credit lines are insufficient, i.e. ν > c + s. In this case, equity
holders receive a low residual claim, Θ. At t=2, the actual cost-income ratio, k,
and net working capital, w, become public knowledge. We define net working
capital, w, as accounts receivable, AR, minus accounts payable, AP, relative to
total assets (i.e. w = AR−APA ). Equity holders receive their residual claim if they
can repay debt and interest. Otherwise, debt holders invoke an insolvency default.
4Considering a concave production function does not alter findings.
7
With the CCC ending, all inflows and costs are realized so net working capital is
zero. The residual claim refers to cash-flows, fixed assets after depreciation, and
cash holding.
To distinguish between transaction and precautionary motives, we use two
models. Model I assumes that cash flows over the whole period are certain, i.e.
A-6 does not apply. The only uncertainty stems from the timing of cash flows,
whether they occur at t = 1 or at t = 2. Model II permits uncertain cash flows. The
following example illustrates Model I. We assume that the firm has total revenues,
REV1 + REV2, of 120 and total operating costs, COS 1 + COS 2, of 100, excluding
costs related to working capital management. Table 1 shows the timing of cash
flows. Note that cash flows over the whole period are certain, and the cost-income
ratio k refers to total costs divided by total revenues over the whole period. The
discount rate is zero based on A-4. In our example, the cost-income ratio is k =
100120 = 5
6 .
Table 1: Illustration of the model
Time t = 0 t = 1 t = 2 periodRevenues REV - 60 60 120Costs COS - 70 30 100Net flow - −10 30 20Accounts receivable AR - 60 - -Accounts payable AP - 30 - -Net working capital - 30 - -Liquidity need - 10 - -
If cash outflows outweigh cash inflows at t=1 a short-term liquidity need la-
beled LIQ arises. Equation (1) defines the short-term liquidity need and links it
8
to cash flows and net working capital defined as accounts receivable, AR minus
accounts payable AP. We then express the liquidity need in terms of total as-
sets labeled ν. Obviously, AP1 = COS 2 and AR1 = REV2. Net working capital
w = AR1−AP1A refers to deferred cash flows.
LIQ = COS 1 − REV1
= (COS 1 + COS 2 − AP1) − (REV1 + REV2 − AR1)
= COS 1 + COS 2 − (REV1 + REV2) + (AR1 − AP1)
⇔LIQ
A= ν = w − T (1 − k) (1)
By definition, the short-term liquidity need, LIQ, refers to cash outflows and
inflows at t = 1, which are equal to net working capital, w, minus total net cash
flows over the whole period. Note that equity holders observe cash inflows, REV1,
and outflows, COS 1, at t=1; hence, the short-term liquidity need LIQ = COS 1 −
REV1 is known. Only at t=2, the actual cost-income ratio, k, and net working
capital, w, are known and thus the underlying cause for the short-term liquidity
need can be understood.
Apart from equation (1), we suppress subscripts for the three periods. First,
A-4 applies so that the actual timing of the cash flow is not relevant for deriving
net present values. Second, some variables such as the liquidity need, ν, and net
working capital, w, only occur at t=1. Third, some variables refer to the whole
period such as the cost-income ratio, k.
9
2.2. Model I: The transaction motive
To identify the transaction motive, we deactivate A-6 making the cost-income
ratio certain (k = k), implying no cash-flow and insolvency risk. Consequently,
debt holders provide short-term funding at the risk-free rate (r = r f ).5 From
A-5 and equation (1), the short-term liquidity need is uniformly distributed with
ν ∼ U(ν, ν) with ν = w− T (1− k) and ν = w− T (1− k)). Furthermore, we assume
that c ≥ 0 ≥ w − T (1 − k). Equity holders maximize the expected utility UE(c)
expressed in terms of total assets selecting optimal cash holding.
UE(c) = (1 − c)T (1 − k) − L(1 + i) − (1 − c)r
ν∫c
ν − cw − w
dν + (1 − c)(1 − l) + c
(2)
Equation (2) describes equity holders’ expected residual claim; they receive
total cash flows, (1 − c)T (1 − k), repay long-term debt and interest, L(1 + i), and
cover expected interest payments for using the credit line. Firms resort to short-
term debt only if the short-term liquidity need exceeds cash holding, captured by
the partial expected value in (2).6 Short-term debt can be repaid since net working
capital is zero at t=2. The residual claim also includes fixed assets after depreca-
tion and initial cash holding. Cash used to fund the short-term liquidity need is
5For maximum short-term debt s and a firm without cash holding, sr f + L(1 + r f ) ≤ (1 −k)T implies no insolvency risk. This is the same condition as in proposition (1.3) in Jaffee andModigliani (1969).
6We refer to Gruner and Zoller (1997), Landsman and Valdez (2005) and Winkler et al. (1972)for mathematical properties.
10
repaid at t=2. Equation (2) reveals the two effects of cash holding, being a buffer
for short-term liquidity needs and determining the exposure to risk and reward
through reducing invested capital. Theorem 1 derives optimum cash holding, and
Appendix A provides a proof.
Theorem 1. (Transaction Cash) Equity holders select cash holding based on the
transaction motive, c∗T , since they face an uncertain short-term liquidity need due
to exogenous shocks in net working capital, w.
c∗T =1 + 2ν −
√(1 − ν)2 + 6
r (T (1 − k) − l)(w − w)
3
Transaction cash is strictly positive, c∗T > 0, if the following condition holds.
r > rmin =(T (1 − k) − l)(w − w)
12ν
2+ ν
Theorem 1 describes the trade-off between investing in fixed assets and cash
holding, reflecting the transaction motive. Obviously, if net return on fixed assets
is negative, T (1− k)− l < 0, equity holders do not invest in fixed assets, and there
is no need for short-term financing. Transaction cash, c∗T , only occurs if interest
rates, r, charged for using the credit line, are sufficiently high. Cash holding
has two effects. It reduces expected costs of short-term borrowing and it reduces
investment in fixed assets. The latter leads to a loss of net return on fixed assets,
11
but also limits exposure to short-term liquidity shocks. This trade-off between
cash holding and fixed assets is the primary difference compared to other models.7
2.3. Model II: The precaution motive
To explore precautionary cash holding, we activate A-6 and assume an un-
certain cost-income ratio, k. Our approach differs from extant research in that
cash-flows are only partly random since firms can modify cash-flow risk through
cash holding (i.e., restricting the amount invested in fixed assets). So risk is mul-
tiplicative and not additive as in Acharya et al. (2012). The cost-income ratio
is uniformly distributed with k ∼ U(0, k) (see A-6). The cost-income ratio is
independent from net working capital (w) (see A-7). Uncertain cash-flows im-
ply insolvency risk, π, so debt holders might not be willing to offer an unlimited
credit line, s, for a given cost of debt, r. Thus, the model considers the possibility
of liquidity default at the interim point t=1 if the actual short-term liquidity need
exceeds cash holding plus the credit line, ν > s + c. In the case of liquidity de-
fault, shareholders receive Θ. Later we set Θ = 0 to simplify the model. Based
on equation (1) and A-7, Lemma 1 derives the density function of the short-term
liquidity need fν(ν).
Lemma 1. fν(ν) is the convolution of fx(x) and fy(y), where x = w − T ∼ U(w −
T,w − T ) and y = T k ∼ U(0,Tk). Assuming that operating costs relative to total
assets are smaller than the range of net working capital (i.e. Tk < w−w) provides
7Gryglewicz (2011) assumes debt and equity are selected such that a firm reaches desired cashholding and investment; a trade-off does not occur.
12
the following result based on Killmann and von Collani (2001).
fν(ν) =(
fx ∗ fy
)(ν) =
0 if ν < w − Tν−w+T
(w−w)Tkif w − T ≤ ν < w − T + Tk
1w−w if w − T + Tk ≤ ν < w − T
−ν+w−T+Tk(w−w)Tk
if w − T ≤ ν ≤ w − T + Tk
0 if ν > w − T + Tk
To evaluate expected costs of short-term finance, we need to consider thresh-
olds of fν based on Lemma 1 that are positive because only if ν > 0 cash outflows
outweigh inflows at t=1, creating a financing need. Assuming that w < T , i.e.
the maximum net working capital relative to total assets is less than the capital
turnover, which seems to be plausible for most firms, we can determine the ex-
pected costs of short-term finance. Furthermore, we assume that liquidity risk
exists, implying 0 ≤ s < w− T + Tk, so that w− T < c ≤ ν ≤ c + s < w− T + Tk.
s+c∫c
(ν − c) fν(ν)dν =
s+c∫c
−ν + w − T + Tk
(w − w)Tkdν
=1
(w − w)Tk
[−
12ν2 + wν − T (1 − k)ν
]c+s
c
=−1
2 s2 − cs + ws − T (1 − k)s
(w − w)Tk(3)
Lemma 2 determines the critical cost-income ratio k∗ for insolvency default by
setting equation (2) equal to or less than zero so that the entity value is insufficient
13
to pay debt holders. We replace the upper limit of the short-term liquidity need, ν,
with s + c, since this is the maximum amount available for short-term finance, the
credit line and cash holding.
Lemma 2. The critical cost-income ratio k∗ ∈ [0, k] for insolvency default is
k∗ = 1 −L(1 + i) + (1 − c)r
∫ s+c
c(ν − c) fν(ν)dν − 1 + l − lc
(1 − c)T
= 1 +1 − L(1 + i)
(1 − c)T−
r(−1
2 s2 − cs + ws − T (1 − k)s)
(w − w)T 2k−
lT
Using Lemma 2 and A-6, we define insolvency risk π.
π = 1 −
k∗∫0
1
kdk = 1 −
k∗
k(4)
Differentiating the critical cost-income ratio with respect to cash holding re-
veals the impact of cash holding on the critical cost-income ratio, which drives
insolvency risk. Lemma 3 summarizes the partial impact of cash holding on in-
solvency risk, and Appendix B provides a proof.
Lemma 3. Cash holding increases the critical cost-income ratio, k∗, reducing
insolvency risk, π, if financial leverage is below the critical level, Lmax.
∂π
∂c< 0 if L <
11 + i
+rs(1 − c)2
(w − w)Tk(1 + i)= Lmax
Cash holding is selected after the credit line is announced. Equity holders can
regard the credit line as a known parameter in their optimization problem. Since
14
it is a two-stage decision problem, we analyze the second stage first, and section
2.4 deals with the first stage. Equity holders do not consider insolvency risk, π,
directly since they are risk-neutral with a linear payoff profile. The investment de-
cision is based solely on the expected cost-income ratio, E(k).8 Modifying equa-
tion (2) yields the following expected utility, which incorporates the possibility of
liquidity default.9
UE(c) =[(1 − c)T (1 − E(k)) − L(1 + i) + 1 − l + lc
] 1 −w−T+Tk∫s+c
fν(ν)dν
− (1 − c)r
s+c∫c
(ν − c) fν(ν)dν + Θ
w−T+Tk∫s+c
f (ν)dν (5)
Differentiating equation (5) with respect to cash holding leads to Theorem 2.
Appendix C provides a detailed proof.
Theorem 2. (Precautionary cash holding) Precautionary cash holding occurs
only if financial constraints are applied, s + c < ν. Optimum precautionary cash
holding is
c∗P =2α(ν − s) + α + γ + 2rs −
√(2α(ν − s) + α + γ + 2rs)2
− 6αΩ
3α
8No investment occurs if E(k) > k∗.9The partial expected value for short-term financing costs refers to the conditional expected
value times the probability that the condition occurs (i.e., no liquidity default).
15
where α = T(1 − k
2
)− l > 0, β = (w − w)Tk > 0, γ = 1 − L(1 + i) > 0 and
Ω = α2 (ν − s)2 + (ν − s)(α + γ) + rs
(ν − 1
2 s + 1)− αβ > 0 are independent from c.
Theorem 2 shows that precautionary cash holding occurs if debt holders im-
pose financial constraints such that s + c < ν. By restricting access to finance,
debt holders force equity holders into liquidity risk. To mitigate liquidity risk,
equity holders select precautionary cash holding, which reduces insolvency risk.
Precautionary cash holding can be positive even if the interest rate on short-term
finance is equal to zero, in contrast to cash holding motivated by the transaction
motive. Based on Theorem 1, the price mechanism is unable to reduce demand
for short-term financing as long as interest rates are not sufficiently high. The-
orem 2 confirms that selecting optimum precautionary cash holding maximizes
shareholder value and is therefore a rational choice for equity holders. The fol-
lowing section addresses the question of whether credit rationing, which leads to
precautionary cash holding, is also a rational choice for the debt holder.
2.4. The debt holders perspective and credit rationing
After discussing equity holders’ cash holding decision for a given credit line s,
we turn to the first stage, optimum selection of the credit line s. If a firm survives,
the debt holder receives a fixed payment of the principal and interest earned at
t = 2. If the firm is insolvent, the debt holder receives only a fraction of the claim,
0 < ε < 1, leading to a concave payout structure. The debt holder has an incentive
to influence insolvency risk, π. Shown in Theorem 2, restricting access to short-
term financing, s + c < ν, forces a firm to hold cash, which reduces insolvency
16
risk (see Lemma 3). We assume that the total short-term lending capacity in terms
of total assets denoted s is sufficient to fund short-term liquidity needs, s > ν.
Furthermore, the debt holder provides short and long-term debt.
The debt holder maximizes the utility function, UD(s), by selecting optimum
credit line, s. The debt holder has no market power, and thus is a price-taker. In
line with A-4, we set the risk-free rate equal to zero, r f = 0.
UD(s) = (s(1 + r) + L(1 + i)) (1 − π) + (s + L) πε + s − s (6)
We explore two cases: credit rationing does not occur, s ≥ ν (CASE 1), and
credit rationing occurs, s < ν (CASE 2). In CASE 1, precautionary cash holding
does not exist and access to short-term financing is sufficient to cover the short-
term liquidity need; hence, there is no liquidity default. Theorem 2 does not apply,
and as a consequence, the credit line does not influence cash holding or insolvency
risk, ∂π∂s = 0. Equity holders are not price sensitive as long as r < rmin (see Theorem
1). In fact, increasing the cost of debt increases insolvency risk since the critical
cost-income ratio k∗ declines, ∂π∂r > 0. The debt holder maximizes (6), and since
UD(s) is linear in s, the optimum credit line follows a bang-bang solution. Credit
rationing does not occur, i.e. s∗ = s, as long as cost of debt compensates for
expected default risk, captured in condition (7). Since insolvency risk π depends
on cost of debt, there is no guarantee that (7) holds.
r ≥π(r)(1 − ε)
1 − π(r)(7)
17
CASE 2 considers the possibility of credit rationing through which a debt
holder can influence insolvency risk (see Theorem 2). If r < rmin, transaction
cash is zero based on Theorem 1, and Lemma 3 confirms that cash holding re-
duces insolvency risk. In this case, we deduce from Theorem 2 that ∂π∂s > 0.
Hence, the debt holder considers her impact on insolvency risk when selecting the
credit line. Under certain conditions derived in Appendix D, credit rationing is
inevitable leading to Theorem 3.
Theorem 3. (Credit Rationing) Credit rationing occurs if the following inequality
holds, where Eν0(ν) denotes the partial expected value of the short-term liquidity
need for positive values. In this case, a cost of debt, r satisfying the debt holder’s
participation constraint does not exist.
Eν0(ν)
Tk>
(1 − π)2
1 − ε
Theorem 2 and Lemma 3 show that credit rationing operates by providing in-
centives to equity holders to increase cash holding, which lowers insolvency risk.
Theorem 1 highlights that equity holders are not price sensitive if the cost of debt
is lower than rmin. Therefore, higher cost of debt does not lower insolvency risk
but increases it. The latter point is important to demonstrate that credit rationing
occurs when condition (7) is violated. Theorem 3 formalizes the condition and
illustrates that firms with high expected liquidity needs (E(ν)) and insolvency risk
(π) likely face credit rationing. Credit rationing is a rational choice for the debt
holder.
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3. Discussion of assumptions and robustness checks
This section highlights robustness checks by relaxing model assumptions.
Some assumptions such as A-1, A-2 and A-3 are important for limiting the scope
of the theory, focusing our attention on short-term decisions taken by shareholder
value maximizing decision makers. Risk-neutrality is convenient as it eliminates
the need to specify a precise utility function, capturing decision makers appetite
for risk. A-4 is plausible given the short-term view and simplifies the model.
Introducing discount rates does not add additional insights. Conceptually, there
might be a problem as cost of capital, if we include short-term finance, changes
from t=0 to t=2.
Using uniform distributions such as in A-5 and A-6 is plausible if little is
known about the risks the firm faces, except for worst case scenarios. In addition,
uniform distributions allow closed-form solutions. Given our definition of the
cost-income ratio, which excludes costs related to working capital management,
independence suggested by A-7 between the cost-income ratio and net working
capital is plausible. This assumption can be relaxed; however, at a high cost as
Lemma 1 does not apply. A-8 is consistent with A-4 and in line with the short-
term perspective.
In reality, firms have some control over their net working capital so that A-
5 can be relaxed. For instance, accounts receivable can be converted into cash
using factoring. Yet the extent of a firm’s control depends on various factors,
e.g. negotiating power and market structure. Therefore, firms can affect liquid-
ity risk directly. This causes problems as outlined by Acharya et al. (2013) as
19
’illiquidity transformation’ can trigger revocations, making the credit line revoca-
ble. Modeling active working capital management (e.g. trade credit or inventory
management) is beyond the scope of the paper.
4. Empirical evidence
4.1. Data and definition of variables
This section illustrates the relevance of the theory by extending Bates et al.
(2009). The sample consists of constituents of the S&P500 from 1990 to 2011.
Companies included in the S&P500 at any point during the investigation period
were considered to avoid survivorship bias. Data on lines of credit (s) are un-
available before 1998, expect for 12 firm-year observations in 1997 and fewer
in previous years. Hence, we restrict the analysis to the period 1998 to 2012.
Bloomberg provides data on the 930 companies, resulting in 9,332 firm-year ob-
servations of cash holding. Due to missing data concerning explanatory variables,
the maximum number of observations in regressions was 5,405.
To re-estimate the models in Table III (p. 2001/2002) reported by Bates et al.
(2009), we used the same variables. Cash holding was measured relative to total
assets (c) or as the log of cash holding divided by total assets (Log(Cash/Assets)).
The first-difference in cash ratios indicated year-by-year changes (Changes). We
included lagged cash ratios (Lag dcash) or lagged changes in cash ratios (Lag
cash). We also considered volatility of cash-flows in the industry (Sigma), market-
to-book, firm size (Size), cash-flows relative to total assets (Cash-flow assets),
net working capital relative to total assets (NWC assets), capital expenditure
20
(CAPEX), financial leverage (Leverage), R&D relative to sales (RD sales), a
dummy variable for dividends (Div dummy), and acquisition activity (ACQ C).
Acquisition activity refers to net cash paid for acquisitions relative to total assets.
We deviate from Bates et al. (2009) in that we use nominal values to determine
the log of total assets. Using real values is less common in the literature.
Based on our theoretical model, we derive two propositions in line with The-
orems 1 and 2. Proposition 1 focuses on the trade-off between cash holding and
investment in fixed assets, whereas Proposition 2 explores the impact of credit
rationing.
Proposition 1. There is a trade-off between cash holding and investing in fixed
assets. Hence, net return on fixed assets has a negative and cost of debt has a
positive partial impact on cash holding.
Proposition 2. Credit rationing leads to precautionary cash holding. Accord-
ingly, the credit line and uncertainty in accessing finance have a negative partial
impact on cash holding. Expected liquidity shortages increase precautionary cash
holding.
To test our propositions, we added the following variables: expected net return
on fixed assets (return) as defined in the theoretical model (i.e., E(T (1− k)− l)),10
cost of bank finance (r),11 and credit line (s). The model also incorporates the
variation coefficient of the credit line as a proxy for the uncertainty of access to
10We regressed net return on assets on lagged values and controlled for industry fixed-effects.Predicted values refer to expectations.
11We did not distinguish long- and short-term interest rates due to lack of data.
21
short-term bank financing (vc s). Variable E shortage measures the expected liq-
uidity shortage based on past fluctuations in net working capital. The expected
liquidity shortage refers to the probability that the short-term liquidity need ex-
ceeds the credit line, assuming a normal distribution (Prob(ν > s) = 1 − F(s)).
This can be interpreted as a measure for credit rationing since Prob(ν > s) > 0⇒
s < ν (ν ∈ [ν, ν]). All variables were winsorized at 1% and 99% percentiles to
mitigate the impact of outliers.
4.2. Empirical findings
Table 2 shows descriptive statistics for all variables. The median and mean
of expected net return on fixed assets (return) exceeded the median and mean
of cost of debt (r). Hence, Theorem 1 suggests that cash holding based on the
transaction motive is unlikely since cost of debt is low. On average, firms had a
credit line of 12% of total assets in comparison to average cash holding of 10%
of total assets. Hence, credit lines were substantial. The probability that short-
term liquidity needs exceeded the credit line was, on average, 9% (E shortage).
Therefore, cash holding might be needed to fill the liquidity gap.
(Insert Table 2)
(Insert Figure 2)
Figure 2 plots average cash ratios (c) and credit lines (s) over time, reveal-
ing a steady decline in credit lines and pronounced increase in cash holding. It
22
is conceivable that cash holding serves as a substitute for credit lines or that pre-
cautionary cash holding is needed to address short-term liquidity risk. Figure 2
indicates that liquidity shortfalls (E shortage) have stabilized over time.
Table 3 re-estimates models applied by Bates et al. (2009). All specifica-
tions refer to fixed-effects models, and Table 3 reports various measures of R-
squared (i.e., within, between, and overall), commensurate with fixed-effects mod-
els. Standard errors were robust to clustering to account for heteroskedasticity and
autocorrelation. Most findings accorded with Bates et al. (2009). In line with the
fixed-effects regression in Table III on p. 2002 reported by Bates et al. (2009),
we confirmed that market-to-book and cash-flow to assets have a positive partial
influence on cash ratios. We also found a negative coefficient for net working
capital relative to total assets and CAPEX. Leverage is only significant if we con-
sider the log cash ratio as a dependent variable. In most specifications, Bates et al.
(2009) report a negative impact of firm size, consistent with our findings. The
only surprising finding was the positive effect of acquisition. Bates et al. (2009)
use the same definition and argue that more cash outflows to fund acquisitions
reduce cash holding. In contrast, we revealed that acquisition led to an increase in
the stockpile of cash, a finding in line with Graham and Harvey (2001), Harford
et al. (2008), and Harford (1999).
(Insert Table 3)
Table 4 extends the model by incorporating additional variables suggested by
the theory, using cash ratios (c) as a dependent variable. The first model (A1)
combines all additional explanatory variables, and subsequent models highlight
23
individual impacts. As predicted, credit lines (s) had a negative effect on cash
holding, shown in models (A1) and (D1). Expected liquidity shortages did not
influence cash holding. By construction, our measure of liquidity shortages cor-
relates negatively with credit line.12 Model (F1) considered only E shortage, but
failed to indicate a significant impact. The variation coefficient of credit lines
(vcs) did not explain cash ratios as reported in models (A1) and (E1).
(Insert Table 4)
Using the log cash ratio as a dependent variable led to similar results. Ta-
ble 5 confirms a negative partial impact of credit line in models (A2) and (D2).
In contrast to regressions with the cash ratio as a dependent variable (see Table
4), expected net returns on fixed assets (return) had a negative influence on cash
holding, as predicted by Proposition 1.
(Insert Table 5)
Finally, the decline in credit lines explains the change in cash ratios (see mod-
els (A3) and (D3) in Table 6), supporting Proposition 2. The coefficient of ex-
pected net returns on fixed assets (return) was negative. The latter findings are
consistent with a trade-off between holding-cash and investing in fixed assets,
confirming Proposition 1.
(Insert Table 5)
12The correlation coefficient was -0.531.
24
Variables suggested by the theory improve the empirical models used by Bates
et al. (2009). In fact, a likelihood ratio test rejects restricting model (A1) to the
standard model used by Bates et al. (2009) with LR chi2(6) of 64.29 and p-value
of 0.000. Proportion 1 and 2 can be confirmed. In particular, empirical findings
point to precautionary cash holding and credit rationing as driving forces behind
the recent increase in cash holding.
5. Conclusion
This paper develops a theory of operational cash holding and explores the links
between short-term liquidity needs due to delayed payments, access to short-term
finance, liquidity risk and insolvency risk. Operational cash holding aims to fund a
firm’s daily operations (Lins et al., 2010), and hence the paper takes a short-term
view in the spirit of the literature on the cash conversion cycle (Gitman, 1974;
Richards and Laughlin, 1980). In contrast to Acharya et al. (2012), Almeida et al.
(2004), Denis and Sibilkov (2010), Denis and Sibilkov (2010), and Gryglewicz
(2011) among others, the theory permits endogenous financial constraints. Cash
holding influences cash-flow risk by restricting the amount invested in fixed assets,
which leads to a multiplicative understanding of risk as opposed to additive risk
(Acharya et al., 2012). Consequently, cash holding is a buffer - as in established
theories - but also a lever, moderating cash-flow risk.
Model I demonstrates the trade-off between cash holding and investing in fixed
assets. Theorem 1 reveals that firms are not price sensitive if cost of debt is be-
low the critical value, rmin. Thus, the price mechanism fails to dampen demand
25
for short-term finance. By introducing uncertain cash-flows, Model II shows that
cash holding reduces insolvency risk (see Lemma 3). Since equity holders are
risk-neutral and have a linear payoff profile, they do not internalize their influ-
ence on insolvency risk. Theorem 2 illustrates that credit rationing exposes equity
holders to liquidity risk, and they respond by holding precautionary cash, which
lowers insolvency risk. This mechanism does not require market frictions such
as information asymmetry; it simply stems from the payoff structure of debt and
equity holders. Theorem 3 illustrates that credit rationing is inevitable if the par-
ticipation constraint (7) is violates. Theorem 3 generalizes to any setting in which
insolvency risk depends on cost of debt, and cost of debt is the solution of the
fixed point iteration rn+1 = g(rn) (n = 1, 2, ...). If g(rn) is not a contraction map-
ping, there is no cost of debt that compensates for insolvency risk, causing credit
rationing. The theory contributes to earlier literature on credit rationing in the
absence of market frictions (Baltensperger, 1978; Jaffee and Modigliani, 1969).
Two propositions capture theoretical predictions and are tested using U.S. data
from 1998 to 2012. Empirical findings confirm theoretical predictions. In partic-
ular, the steady decline in credit lines has contributed to the increase in cash hold-
ing. We stress that our theoretical model should be re-tested by other researchers.
26
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Figure 1: Model structure
t=0Start of CCCKnown parameters:T , l, r, i and L
Net working capital:w ∼ U(w,w)
Cost-income ratio:k ∼ U(0, k)
Debt holder:decides about s for given rexpected insolvency risk π
Equity holder:decides about c for known sexpected cost-income ratio E(k)expected short-term liquidity need E(ν)
t=1Interim pointNew information:actual ν = COS 1 − REV1
Outcomes:(a) s + c ≥ νno liquidity problem(b) s + c < νliquidity default→ Θ
t=2End of CCCNew information:actual k and ν
Outcomes:(a) k < k∗
No insolvency(b) k ≥ k∗
Insolvency
31
Figure 2: Average cash holding (c), liquidity shortages (E shortage) and credit lines over time (s)
1998 2000 2002 2004 2006 2008 2010 2012
8
10
12
14
16
18
year
Perc
ent(
%)
cs
E shortage
The figure depicts annual sample averages of cash ratios (c) and credit lines (s) in percent of totalassets. It also shows the average expected liquidity shortage (E shortage).
32
Table 2: Descriptive statistics
The sample includes all firm-year observations from 1998 to 2012. The vari-ables contain cash ratios (c), the expected return on investment (return), costof debt (r), the credit line (s), the variation coefficient of the credit line (vc s),expected liquidity shortage (E shortage), standard deviation of cash flows inindustry (Sigma), market-to-book ratios, firm size, cash flows relative to assets(Cash flow assets), net working capital relative to assets (NWC assets), capitalexpenditure, leverage, R&D spending relative to sales (RD sales), dividendand acquisition dummies.
mean sd p25 p50 p75 min maxc 0.10 0.10 0.03 0.07 0.14 0.00 0.48return 0.11 0.05 0.07 0.10 0.14 -0.07 0.28r 0.08 0.12 0.05 0.06 0.08 0.00 1.00r2 0.02 0.11 0.00 0.00 0.01 0.00 1.01s 0.12 0.10 0.05 0.10 0.17 0.00 0.52vc s 0.57 0.29 0.35 0.54 0.82 0.00 1.00E shortage 0.09 0.14 0.00 0.01 0.13 0.00 0.50Sigma 0.06 0.03 0.03 0.05 0.07 0.00 0.23Market to book 4.07 4.05 1.86 2.91 4.55 0.52 27.68Size 8.87 1.25 7.97 8.78 9.74 3.85 11.64Cash flow assets 0.11 0.07 0.07 0.11 0.15 -0.10 0.34NWC assets 0.04 0.08 0.00 0.05 0.09 -0.22 0.33CAPEX 0.05 0.04 0.02 0.04 0.06 0.00 0.26Leverage 0.23 0.14 0.13 0.22 0.32 0.00 0.64RD sales 0.06 0.08 0.00 0.02 0.07 0.00 0.50DIV dummy 0.67 0.47 0.00 1.00 1.00 0.00 1.00ACQ C 0.00 0.00 0.00 0.00 0.00 0.00 0.02
33
Table 3: Standard models with fixed-effects
The sample includes all firm-year observations from 1998 to 2012. The mod-els are estimated using OLS with fixed-effects. Standard errors are robustto clustering to account for heteroskedasticity and autocorrelation. The firstmodel corresponds to the fixed-effects specification in Table III used in Bateset al. (2009). The second model refers to log cash ratios, and the third modelexplains the changes in cash ratios.
Cash/Assets Log(Cash/Assets) ChangesLag dcash -0.061∗∗
Lag cash -0.443∗∗∗
Sigma -0.008 -0.541 0.043Market to book 0.001∗ 0.008 0.000Size -0.016∗∗∗ -0.044 -0.010∗∗∗
Cash flow assets 0.152∗∗∗ 1.431∗∗∗ 0.169∗∗∗
NWC assets -0.273∗∗∗ -2.614∗∗∗ -0.136∗∗∗
CAPEX -0.430∗∗∗ -3.985∗∗∗ -0.312∗∗∗
Leverage -0.020 -1.032∗∗∗ 0.023RD sales -0.005 0.511 0.036DIV dummy 0.002 -0.036 0.001ACQ C 0.745∗∗ 9.092∗∗∗ 0.012r2 w 0.068 0.066 0.317r2 b 0.204 0.300 0.036r2 o 0.153 0.221 0.165N 5405 5394 5280∗ p < 0.05, ∗∗ p < 0.01, ∗∗∗ p < 0.001
34
Table 4: Models with Cash/Assets as depedent variable
The sample includes all firm-year observations from 1998 to 2012. The mod-els are estimated using OLS with fixed-effects. Standard errors are robust toclustering to account for heteroskedasticity and autocorrelation. In line withmodel one in Table 2, the cash ratio is the dependent variable. Based on theo-retical considerations, the expected return on investment (return), cost of debt(r), the credit line (s), the variation coefficient of the credit line (vc s), and ex-pected liquidity shortage (E shortage) act as additional explanatory variables.
[A1] [B1] [C1] [D1] [E1] [F1]return 0.025 -0.058r 0.001 -0.003s -0.138∗∗∗ -0.130∗∗∗
vc s 0.011 0.002E shortage -0.014 0.023Sigma 0.005 -0.028 -0.018 0.008 -0.002 0.007Market to book 0.001 0.001∗ 0.001∗ 0.001 0.001∗ 0.001Size -0.009∗ -0.015∗∗∗ -0.008 -0.020∗∗∗ -0.020∗∗∗ -0.019∗∗∗
Cash flow assets 0.170∗∗∗ 0.169∗∗∗ 0.170∗∗∗ 0.149∗∗∗ 0.148∗∗∗ 0.129∗∗∗
NWC assets -0.172∗∗ -0.245∗∗∗ -0.228∗∗∗ -0.267∗∗∗ -0.283∗∗∗ -0.295∗∗∗
CAPEX -0.396∗∗∗ -0.396∗∗∗ -0.392∗∗∗ -0.428∗∗∗ -0.451∗∗∗ -0.442∗∗∗
Leverage -0.011 -0.020 -0.036 0.005 -0.006 -0.007RD sales -0.078 -0.037 0.007 -0.051 -0.014 -0.061DIV dummy 0.005 0.001 0.000 0.002 0.007 0.008ACQ C 0.778∗∗ 0.744∗∗ 0.659∗ 0.883∗∗ 0.877∗∗ 0.748∗∗
r2 w 0.086 0.062 0.064 0.087 0.072 0.071r2 b 0.125 0.177 0.229 0.180 0.153 0.100r2 o 0.109 0.131 0.147 0.156 0.133 0.090N 3053 5344 4487 3809 4472 3665∗ p < 0.05, ∗∗ p < 0.01, ∗∗∗ p < 0.001
35
Table 5: Models with Log(Cash/Assets) as dependent variable
The sample includes all firm-year observations from 1998 to 2012. The mod-els are estimated using OLS with fixed-effects. Standard errors are robust toclustering to account for heteroskedasticity and autocorrelation. In line withmodel two in Table 2, the log cash ratio is the dependent variable. Basedon theoretical considerations, the expected return on investment (return), costof debt (r), the credit line (s), the variation coefficient of the credit line (vcs), and expected liquidity shortage (E shortage) act as additional explanatoryvariables.
[A2] [B2] [C2] [D2] [E2] [F2]return 0.024 -0.945∗∗
r -0.006 -0.019s -2.013∗∗∗ -1.704∗∗∗
vc s 0.058 -0.008E shortage -0.372 0.141Sigma -0.037 -0.622 -0.643 -0.350 -0.321 -0.314Market to book 0.006 0.008 0.011 0.005 0.010∗ 0.004Size -0.025 -0.029 0.024 -0.082 -0.061 -0.042Cash flow assets 1.865∗∗∗ 1.647∗∗∗ 1.654∗∗∗ 1.642∗∗∗ 1.421∗∗∗ 1.523∗∗∗
NWC assets -1.839∗∗∗ -2.434∗∗∗ -2.569∗∗∗ -2.253∗∗∗ -2.544∗∗∗ -2.619∗∗∗
CAPEX -4.326∗∗∗ -3.555∗∗∗ -4.131∗∗∗ -3.843∗∗∗ -4.054∗∗∗ -4.233∗∗∗
Leverage -0.934∗∗∗ -1.075∗∗∗ -1.301∗∗∗ -0.739∗∗ -0.797∗∗∗ -0.949∗∗∗
RD sales 0.440 0.175 0.542 0.580 0.637 0.687DIV dummy 0.000 -0.034 -0.084 -0.001 0.028 0.024ACQ C 9.437∗∗∗ 9.307∗∗∗ 9.150∗∗∗ 9.368∗∗∗ 9.264∗∗∗ 8.696∗∗∗
r2 w 0.095 0.066 0.077 0.087 0.056 0.063r2 b 0.244 0.272 0.272 0.299 0.257 0.240r2 o 0.188 0.200 0.178 0.243 0.201 0.198N 3048 5333 4476 3804 4466 3660∗ p < 0.05, ∗∗ p < 0.01, ∗∗∗ p < 0.001
36
Table 6: Models with change in cash ratios as dependent variable
The sample includes all firm-year observations from 1998 to 2012. The mod-els are estimated using OLS with fixed-effects. Standard errors are robust toclustering to account for heteroskedasticity and autocorrelation. In line withmodel three in Table 2, the change in cash ratio is the dependent variable.Based on theoretical considerations, the expected return on investment (re-turn), cost of debt (r), the credit line (s), the variation coefficient of the creditline (vc s), and expected liquidity shortage (E shortage) act as additional ex-planatory variables.
[A3] [B3] [C3] [D3] [E3] [F3]return -0.032 -0.102∗∗
r 0.007 -0.001s -0.098∗∗∗ -0.086∗∗∗
vc s 0.000 -0.002E shortage -0.025 0.003Sigma 0.071 0.049 0.060 0.034 0.027 0.035Market to book 0.000 0.000 0.001 -0.000 0.000 -0.000Size -0.009∗∗ -0.008∗∗∗ -0.007∗∗ -0.016∗∗∗ -0.012∗∗∗ -0.014∗∗∗
Cash flow assets 0.192∗∗∗ 0.192∗∗∗ 0.168∗∗∗ 0.168∗∗∗ 0.169∗∗∗ 0.154∗∗∗
NWC assets -0.091∗ -0.121∗∗∗ -0.134∗∗∗ -0.132∗∗∗ -0.143∗∗∗ -0.151∗∗∗
CAPEX -0.317∗∗∗ -0.276∗∗∗ -0.306∗∗∗ -0.329∗∗∗ -0.324∗∗∗ -0.333∗∗∗
Leverage 0.020 0.021 0.008 0.034∗ 0.032 0.025RD sales -0.031 0.007 0.032 -0.019 0.018 -0.018DIV dummy 0.002 0.001 -0.001 -0.000 0.003 0.003ACQ C 0.456 0.065 0.210 0.272 0.177 0.254r2 w 0.376 0.316 0.335 0.345 0.325 0.339r2 b 0.005 0.039 0.011 0.001 0.007 0.000r2 o 0.112 0.164 0.154 0.122 0.147 0.115N 3011 5278 4424 3717 4367 3573∗ p < 0.05, ∗∗ p < 0.01, ∗∗∗ p < 0.001
37
Appendix A. Theorem 1
Proof. We differentiate equation (2) with respect to c and obtain the first-order
condition (FOC).
dUE(c)dc
= −T (1 − k) + l + r
ν∫c
ν − cw − w
dν − (1 − c)r
ν∫c
−1w − w
dν = 0
− T (1 − k) + l +r
w − w
ν∫c
(ν − 2c + 1) dν = 0
− T (1 − k) + l +r
w − w
[12ν2− 2cν + ν −
12
c2 + 2c2 − c]
= 0
32
c2 − (1 + 2ν)c +12ν2
+ ν −(T (1 − k) − l)(w − w)
r= 0
If a liquidity need can arise, i.e. ν = w − T (1 − k) > 0, there will be one
optimal value of cash holding maximizing UE(c). To see that, note that the FOC
is a U-shaped parabola and that the second-order condition implies c < 1+2ν3 to
ensure a maximum, suggesting that the maximizer is the smaller root of the FOC.
Hence, we can state the maximizer, labeled c∗T .
c∗T =1 + 2ν −
√1 + 4ν + 4ν2
− 3ν2− 6ν + 6
r (T (1 − k) − l)(w − w)
3
=1 + 2ν −
√1 − 2ν + ν2
+ 6r (T (1 − k) − l)(w − w)
3
=1 + 2ν −
√(1 − ν)2 + 6
r (T (1 − k) − l)(w − w)
3
38
To ensure that firms hold transaction cash, i.e. dUE(c)dc
∣∣∣c=0
> 0 , the following
inequality has to hold.
r >(T (1 − k) − l)(w − w)
12ν
2+ ν
Appendix B. Lemma 3
Proof. Based on Lemma 2, we simplify the critical cost-income ratio focusing on
terms involving c, where A refers to terms independent from c.
k∗ = A +1 − L(1 + i)
(1 − c)T+
rcs
(w − w)T 2k
Then we determine ∂k∗∂c , i.e. the partial impact of cash holding on the critical
cost-income ratio.
∂k∗
∂c= −
1 − L(1 + i)T (1 − c)2 (−1) +
rs
(w − w)T 2k
=1 − L(1 + i)T (1 − c)2 +
rs
(w − w)T 2k
To ensure that ∂k∗∂c > 0, financial leverage needs to be below a critical level,
39
Lmax.
1 − L(1 + i)T (1 − c)2 +
rs
(w − w)T 2k> 0
1 − L(1 + i) > −rs(1 − c)2
(w − w)Tk
L <1
1 + i+
rs(1 − c)2
(w − w)Tk(1 + i)= Lmax
Appendix C. Theorem 2
Proof. Equation (5) can be simplified using (3) and setting Θ = 0 without loss of
generality.
UE(c) =
(1 − c)T1 − k
2
− L(1 + i) + 1 − l + lc1 −
w−T+Tk∫s+c
fν(ν)dν
− (1 − c)r
−12 s2 − cs + ws − T (1 − k)s
(w − w)Tk(C.1)
40
Then we evaluate the probability of liquidity default∫ w−T+Tk
s+cfν(ν)dν.
w−T+Tk∫s+c
fν(ν)dν =
−12ν
2 + (w − T + Tk)ν
(w − w)Tk
w−T+Tk
s+c
=1
(w − w)Tk
[12
(w − T + Tk)2 − (w − T + Tk)(s + c) +12
(s + c)2]
=(w − T + Tk − s − c)2
2(w − w)Tk(C.2)
Using (C.2) in (C.1) and noting that (w − w)Tk is a constant.
UE(c) =
(1 − c)T1 − k
2
− L(1 + i) + 1 − l + lc 1 − (w − T + Tk − s − c)2
2(w − w)Tk
− (1 − c)r
−12 s2 − cs + ws − T (1 − k)s
(w − w)Tk
∝
(1 − c)T1 − k
2
− L(1 + i) + 1 − l + lc [(w − w)Tk −
12
(w − T + Tk − s − c)2]
− (1 − c)rs(w − T + Tk −
12
s − c)
Using ν = w − T + Tk, we finally obtain
UE(c) ∝(1 − c)T
1 − k2
− L(1 + i) + 1 − l + lc [(w − w)Tk −
12
(ν − s − c)2]
− (1 − c)rs(ν −
12
s − c)
41
Next, we derive the FOC.
dUE(c)dc
=
−T1 − k
2
+ l [(w − w)Tk −
12
(ν − s − c)2]
+
(1 − c)T1 − k
2
− L(1 + i) + 1 − l + lc (ν − s − c)
+ rs(ν −
12
s − c)
+ (1 − c)rs = 0
To simplify the FOC, we introduce three constants that depend on model
parameters but do not depend on c and s so that α = T(1 − k
2
)− l > 0,
β = (w − w)Tk > 0 and γ = 1 − L(1 + i) > 0.
dUE(c)dc
= −α
[β −
12
(ν − s − c)2]
+[(1 − c)α + γ
](ν − s − c)
+ rs(ν −
12
s − c)
+ (1 − c)rs = 0
So that
dUE(c)dc
= −αβ +α
2
((ν − s)2 − 2(ν − s)c + c2
)+ (1 − c)α(ν − s − c) + γ(ν − s − c)
+ rs(ν −
12
s + 1)− 2rsc = 0
42
Then
dUE(c)dc
= −αβ +α
2(ν − s)2 − α(ν − s)c +
α
2c2
+ α(ν − s) − αc − α(ν − s)c + αc2
+ γ(ν − s) − γc − 2rsc + rs(ν −
12
s + 1)
= 0
Finally we obtain
dUE(c)dc
=32αc2 − (2α(ν − s) + α + γ + 2rs) c
+α
2(ν − s)2 + (ν − s)(α + γ) + rs
(ν −
12
s + 1)− αβ = 0
The second-order condition determines an upper bound for optimal cash hold-
ing to ensure that a maximum is reached.
d2UE(c)dc2 = 3αc − (2α(ν − s) + α + γ + 2rs) < 0
⇔ c <2α(ν − s) + α + γ + 2rs
3α= cup
Accordingly, optimal cash holding based on the precaution motive, c∗P is the
smaller root of the FOC. Therefore, there is at most one c∗P ∈ [0, cup] that maxi-
mizes UE(c). Using Ω = α2 (ν − s)2 + (ν − s)(α + γ) + rs
(ν − 1
2 s + 1)− αβ > 0 for
43
the term that does not depend on c, we obtain the maximizer c∗P.
c∗P =2α(ν − s) + α + γ + 2rs −
√(2α(ν − s) + α + γ + 2rs)2
− 6αΩ
3α
It is easy to see that c∗P > 0, which is not guaranteed for transaction cash c∗T
only if the interest rate on short-term finance is in a permitted range (see The-
orem 1). To ensure that a solution exists, the following condition has to hold
(2α(ν − s) + α + γ + 2rs)2− 6αΩ ≥ 0.
Appendix D. Theorem 3
Proof. Credit rationing does not occur (i.e. s∗ = s > z) if and only if condition (7)
holds. The RHS of (7) depends on r, since π(r) with ∂π∂r > 0. We regard (7) as a
simple fixed-point iteration with rn+1 = g(rn) (n = 1, 2, ...), where g(r) =π(r)(1−ε)
1−π(r) .
Starting the iteration with the lowest possible value r0 = r f = 0 and, as long as
π(r0) > 0 ⇒ r1 < g(r0), r has to increase so that (7) holds. Increasing the LHS of
(7) also increases the RHS of (7). To insure that (7) holds, the marginal change on
the LHS has to outweigh the marginal change on the RHS. This is an application
of the contraction mapping theorem, and if g′(r) > 1, g(r) is not a contraction
44
mapping and hence (7) does not hold.
g′(r) =
∂π∂r (1 − ε)(1 − π) + ∂π
∂rπ(1 − ε)(1 − π)2 > 1
⇔∂π
∂r>
(1 − π)2
1 − ε
∂π∂r follows from Lemma 2 and (4) with c = 0 as r < rmin and s = ν as there
is no credit rationing. In addition, the partial expected value for the short-term
liquidity need is positive and can be written as follows.
s+c∫c
(ν − c) fν(ν)dν =
ν∫0
ν fν(ν)dν = Eν0(ν) > 0
Hence, the partial impact of r on the critical cost-income ratio is
∂k∗
∂r= −
1T
Eν0(ν) < 0
Then we obtain dπ∂r .
∂π
∂r= −
∂k∗
∂r1
k=
Eν0(ν)
Tk> 0
Hence
Eν0(ν)
Tk>
(1 − π)2
1 − ε
45